The Questionable Uses of the Term "Probability" in Economics

"I give a 50% chance that Sandy's storm surge will end up flooding a portion of the New York City subway system."
—Dr. Jeff Masters, October 28, 2012 (approximately 24 hours before the expected landfall of the large storm)1

"My personal opinion is that there is no single correct definition of probability. Instead, the word 'probability' means different things in different contexts."

I took note of the blog post quoted above, and in fact began to write this essay, before the outcome of the hurricane was known. Indeed, a portion of the New York subway system was flooded. Does this tell us that Dr. Masters was correct? Had there been a different outcome, with a dry subway system, would that have told us that he was incorrect? It strikes me that there is no outcome that could have proven Dr. Masters' statement false, and this would be the case even had he chosen 10 percent or 80 percent as his numerical figure. In my view, his statement, while useful as a warning, had only faux scientific content.

Dr. Masters used "50% chance" in a way that suggested that "chance" was synonymous with probability. Probability is often used in a scientific context. What I want to argue in this essay is that the term "probability" has several meanings, at least one of which is not scientific. By conflating the scientific and non-scientific usages, economists fall into conceptual traps, most dramatically in theorizing about "rational expectations."

There is a well-known philosophical disagreement about the meaning of probability in the statement, "Flipping a fair coin, the probability of heads is 50 percent." One school of thought says that the 50 percent probability is definitional—it is what we mean by the term "fair coin." We do not need to observe any coin flips in order to make calculations that assume such a 50 percent probability.

Another school of thought says that the 50 percent probability is empirical. We only come to believe that the probability is 50 percent after observing for ourselves or receiving credible reports from others that large sequences of coin flips produce heads 50 percent of the time.

A third school of thought is that probability is subjective. It is a sort of thermometer that measures the likelihood of an event in the mind of the beholder.

My personal opinion is that there is no single correct definition of probability. Instead, the word "probability" means different things in different contexts. This linguistic misfit could be corrected if we were to use three words instead of one. Let the three words be probadef, probastat, and probatherm, respectively.

1. Probadef would be used to signify probabilities that are definitional. Probadef statements are like proofs in mathematics. They can be given as axioms or derived as tautologies. Probadef applies to theoretical calculations about how a fair coin or unloaded dice or a variable following the Gaussian normal distribution would behave.

2. Probastat would be used to signify probabilities that are based on empirical observation under repeatable conditions. Probastat statements are like observations about the natural world. They can be verified by gathering data. Probastat applies to samples, such as the heights and weights in a random sample of 14-year-old males. In principle, if you make a statement based on a sample, I can verify that statement by taking my own observations using a similar sample. Because probastat statements are verifiable, they can be called scientific.

3. Probatherm would be used to signify probabilities that act as a sort of thermometer, measuring the degree of conviction expressed by the individual that an event will occur. These are faux-scientific statements, because they cannot be verified.

When Dr. Masters, the meteorologist, said that he foresaw a 50 percent chance of the storm flooding part of the New York subway system, this was probatherm. Clearly, the 50 percent chance of flooding was not some mathematical tautology, so it was not probadef. It also was not probastat, because the conditions were not repeatable. In fact, elsewhere in his analysis, Dr. Masters, like other meteorologists, called the storm "unprecedented." Thus, the phrase "50% chance" was an example of probatherm, a statement that represents the degree of conviction that this meteorologist had about a prediction that the storm would flood the New York subway system.

I do not believe that anyone misunderstood what Dr. Masters meant by his statement. I believe nobody would confuse what he was saying with an empirical description of repeatable events. Everybody would understand that he was uncertain about the outcome. They would have inferred that he had a high but not overwhelming level of conviction that part of the subway system would be flooded.

Economists have long realized that economic agents, meaning consumers and producers, inhabit a world of uncertainty. In the future, how will various assets be valued? How will different investment projects turn out? What sorts of natural disasters and man-made adversities will we encounter?

Naturally, then, economists proceed to model the behavior of economic agents by assuming that these agents form probability estimates. However, there is an interesting division between economists who tend to treat these probability estimates as probatherm (subjective) and those who treat them as probadef or probastat (objective).

On the objective side, I put the "rational expectations" school of economics. This is represented by a number of Nobel laureates, notably Robert Lucas and Thomas Sargent. Economists who adhere to this approach treat agents as if they derive probabilities from the set of equations in an economic model that the economist stipulates.

On the subjective side, I put John Maynard Keynes. He wrote a treatise on probability, in which he leaned toward a probatherm approach.2 In addition, some of the most memorable ideas in his macroeconomic classic The General Theory of Employment, Interest, and Money, elaborate on probatherm. Entrepreneurs choose on the basis of "animal spirits," not probastat. The stock market is like a beauty contest for newspaper readers, in which in order to win you must guess who everyone else believes the winner ought to be.

What about Keynes' intellectual adversary, Friedrich Hayek? On this issue, I put him on the same side, because Hayek emphasized the local nature of knowledge. Different agents have access to different information sources and may have different approaches to processing information. Therefore, Hayek, like Keynes, would view consumers and entrepreneurs as navigating through uncertainty by using probatherm.

Note that I am far from the first economist to link Hayek, Keynes (and also Frank Knight) in this way. For example, see Roman Frydman and Michael Goldberg's works: Imperfect Knowledge Economics (2007) and the more accessible Beyond Mechanical Markets (2011), both from Princeton University Press.

Frydman and Goldberg point out that the rational expectations approach requires economic probabilities to be objectively knowable, so that economic agents can acquire them. The Keynes-Hayek-Knight view is that probabilities are like consumer tastes or specialized skills, which can differ from individual to individual.

Producers and consumers live in a world of non-repeatable events. Like a meteorologist trying to prognosticate about hurricane Sandy, our decisions are based on probatherm statements, not probadef statements or probastat statements. Treating probabilities as if they were objective is a conceptual error. It is analogous to the conceptual errors that treat value as objective.3 We will be less likely to overstate the robustness of equilibrium and the precision of economic models if we stop conflating subjective degrees of conviction with verifiable scientific concepts of probability.

*Arnold Kling has a Ph.D. in economics from the Massachusetts Institute of Technology. He is the author of five books, including Crisis of Abundance: Rethinking How We Pay for Health Care; Invisible Wealth: The Hidden Story of How Markets Work; and Unchecked and Unbalanced: How the Discrepancy Between Knowledge and Power Caused the Financial Crisis and Threatens Democracy. He contributed to EconLog from January 2003 through August 2012.

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